Itinerary for Family with 7 & 9 Year Old 15 Days - Fly into Dubrovnik / Out of Split

From Dubrovnik’s old Town, Lokrum National Park is just a 15-minute ferry ride away. The island of Mljet with its national park is an 80-minute ferry ride away. You might consider staying a couple nights on Mljet, rather than staying all five nights in Dubrovnik which often sees thousands of cruise ship passengers flood the old city.
Makarska and the beautiful beaches near it, make up the “Makarska Riviera.” With a car, this town can really be a relaxing base for a few days. It’s far less congested than Split, which is Croatia’s second most-populated city. Split also receives cruise ships and ferries that dock at its port every day.

An hour north of Split is the coastal town of Sibenik, which has many beaches accessible from the towns surrounding it. Krka national Park, with its waterfalls and riverside pathways, is just a fifteen-minute drive from Sibenik.

Also look at Trogir to stay instead of Split .... but visit Split
When I took my kids to Croatia they were close to 30 years old. .....
So ...

Read these
https://www.lonelyplanet.com/articles/croatia-with-kids
https://thefamilyconscience.com/the-best-croatia-itinerary-for-families/
https://kamifridayphotography.com/family-travel-to-croatia-with-kids/

With younger children I would look into staying on one of the islands accessible by ferry.
Read about Hvar or Korčula.
I haven’t been to Hvar but I know it is popular, maybe more with young adults
On Korčula I know you can rent bikes, there are many beautiful beaches, and many families with young kids. So I am sure there is more to do than I know about.
There are also water taxis that can take you to small islands to swim and snorkel for the day.
Split is very crowded in the summer and therefore so are the beaches and activities.
You can ferry from Dubrovnik to Hvar or Korčula then to Split, no need for a car.

You really don't need to rent a car unless you select things on the mainland that you want to prioritize--the catamaran stops at several islands en route from Dubrovnik to Split, so three roughly five-day stops would be my preference (selecting one island in the middle). You could also take a car and select islands with car service en route (Korcula is one).
See Jadrolinija (car ferries and catamarans) and Krilo websites for schedules.

Also, what sources are you using to help you choose places? You can do most things on your list in most places, so a lot comes down to just picking what draws you (for me, Lastovo, someone else may be drawn to Brac--no wrong choices!).
We definitely need to know if you want to stay in the most famous places or you would prefer staying elsewhere and doing day trips, that kind of thing. You do not need to spend ten days in the Split area unless you just wanted all of the day trip options from one place--that could get tiring in its own way. With 15 days, four stops would not be too much really--pack light and hop on a catamaran (it's a a wonderfully easy place to travel in my experience, but I do acknowledge it has gotten much more crowded in recent years).

We're all different travelers
I am not a fan of crowds, I like getting to know the culture and feeling the charm and vibe.
Crowds do not bother my husband and he likes to see the famous sites
My kids - just want to go to amusement parks (I know there are none), and they want to see the crystal clear water and try to snorkel
So trying to find a balance for all.
We went to Portugal and because of the crowds, my kids thought it was just like being at a city at home. We stayed in 1 place 3 days and another 4 days and that was too rushed. So trying to make this trip better.
We already booked Dubrovnik for 5 days. So we have 10 days to play with.
1. We want to do some day trips from a next base - national park(s), maybe Zadar?, maybe a day at the Solaris hotel for the kids
2. We'd like to do an (or multiple) islands. Should we base in Split and then do day trips? After Dubrovnik should we stay on an island a couple days, then settle in / near Split?
3. If staying in Split, should we stay in town near Meje? I also found a wonderful cottage in Kaštel Sućurac. Is that too far away?
4. I was thinking of renting bikes at some point - anywhere, again to give kids an activity

1. We want to do some day trips from a next base - national park(s), maybe Zadar?, maybe a day at the Solaris hotel for the kids There are many national parks to choose from, so it depends on which ones you select, but I would stick to the route from Dubrovnik to Split. You all get quiet and something different form home on an island, dad gets some history in Dubrovnik and Split.
2. We'd like to do an (or multiple) islands. Should we base in Split and then do day trips? After Dubrovnik should we stay on an island a couple days, then settle in / near Split? I think an island or two between Dubrovnik and Split makes perfect sense given your description of wanting quiet and outdoor activities.
3. If staying in Split, should we stay in town near Meje? I also found a wonderful cottage in Kaštel Sućurac. Is that too far away? I would want to stay within a walk or taxi ride of the ferry port if you will be doing frequent day trips.
4. I was thinking of renting bikes at some point - anywhere, again to give kids an activity Definitely include a stay on an island so the kids can bike freely--among the vineyards at Lumbarda on Korcula is one option. I am not 100% but I wonder if on Mljet there are bike rentals?